Zero-one laws for polynomials in Gaussian random variables: A simple proof

Zero-one laws for polynomials in Gaussian random variables have already been studied.⁽⁷⁾ They are established here by very simple arguments: Fubini's theorem and the rotational invariance of centered Gaussian measures. The proof is built on the Polarization formula that has received much attention in Refs. 1 and 5. Our point of view derives from the deep work of Borell.⁽²⁾ In a natural way, these results extend to finite-order Gaussian chaos processes.